A standard practice in processing an image from a CCD is to subtract off
the dark current pattern, since it is an unwanted signal that is added
to the data. In order to have the proper dark current reference frames
for this subtraction, we created some exposures with the accelerating
high voltage turned off. This was done throughout the mission, so that
we could monitor changes in this pattern. The dark frames had the
additional benefit of allowing us to identify pixels and columns that
that gave anomalous signals (usually showing up as hot spots).

The dark current of a CCD varies markedly with temperature, as one would
expect. As stated earlier in §3.5, the overall level of this
background is adjusted to remain constant by an automatic change in a
bias in the electronics, before the frame is digitized. However, higher
order effects remain. Dark currents are slightly different from one
pixel to another, and different pixels change their dark currents by
differing amounts when there is a change in temperature. We can see
this effect as a change in the pattern of a random checkerboard at
different times in the mission (as the temperature varies). Also,
certain columns and some broad areas become brighter or darker relative
to the rest of the frame when the temperature changes.

To improve on the quality of a spectral image, it is important to remove
the CCD's dark current pattern. However, it is also important that the
pattern that is subtracted from a picture be one that applies to the
correct temperature. We had no independent measure of the CCD's
temperature, so the correct matching of the pattern had to be done on
the basis of a consistency check between the particular image with a
spectrum and an array of dark current patterns recorded over the range
of operating temperatures.

On the assumption that each pixel has a dark current that varies
linearly with a small change in temperature but with an arbitrary zero
point and slope, one can take any two background images recorded at
different temperatures and, by linear interpolation or extrapolation of
each pixel's trend, reconstruct the entire dark-current pattern at an
arbitrary temperature within our operating range. This simple process
is not very accurate because there is random noise present in each of
the two pictures. However, the noise can be reduced considerably if a
large number of cases is used. By minimizing the squares of the errors
for all pixels, one can place any background image in its proper place
in a sequence of an empirical ``temperature'' parameter q that has
been defined for a whole collection of images. Once this has been done,
the behavior of each pixel can be defined by the best linear fit to all
the measurements taken at different values of q.

The determination of where in the temperature sequence a given
background image lies (i.e., what is its value for q) is relatively
simple because the entire signal is the dark current pattern, aside from
random readout noise and the coherent electronic noise that could be
filtered out (as described earlier in §7.2). For the images
that contain a spectrum, the process is more difficult. The challenge
is to devise a method of temperature matching that is insensitive to the
perturbing influence of the spectrum signal, which in some cases can be
very strong relative to the dark current pattern. As with the
electronic interference discussed in §7.2, the answer again
rests upon taking advantage of the fact that most of the power in the
Fourier transform of the spectrum is restricted to low frequencies. It
follows that the checkerboard pattern from dark current variations
dominates at high frequencies. Thus, it is appropriate to determine the
temperature parameter q by minimizing the squares of the errors in the
Fourier domain, and to do this only at high frequencies.

The analysis took the following form. For the sake of argument, let q
range between 0 and 1. These values are defined to represent the
extremes measured in the background images. If we define the Fourier
transform of the background picture at q=0 to be f(a), let
represent the transform of the picture at q=1 minus the
one at q=0, and designate the spectral image's transform to be f(I),
then the square of the error for the i,jth Fourier component is given
by

(2)

Over an appropriate range of high frequencies in i and j we must
solve for

(3)

which leads to the expression for q,

(4)

For every image with a spectrum I, we evaluated q and then produced
a background corrected version from an appropriate linear
combination of two template pictures a and ,

(5)

When this subtraction is done, the amplitude of the random checkerboard
decreases substantially, and virtually all that remains are other
sources of noise.